SQUID-ON-TIP

CROSS REFERENCE TO RELATED APPLICATIONS

[001] This application claims priority of U.S. Provisional Patent Application Serial No. 62/149,671, filed April 20, 2015, entitled "Fabrication and characterization of multi -junction, multi -terminal Superconducting Quantum Interference Device On a Tip", which is incorporated in its entirety herein by reference.

BACKGROUND

[002] In recent years there has been growing interest in the development of nanoscale magnetic characterization tools in order to address the rapidly evolving fields of nanomagnetism and spintronics.

[003] Scanning micro and nanoscale superconducting quantum interference devices (SQUIDs) are of particular interest in magnetic imaging, because of their high- sensitivity and high bandwidth. The two main technological approaches for the fabrication of scanning SQUIDs are based on planar lithographic methods and on self- aligned SQUID-on-tip (SOT) deposition.

[004] Planar SQUIDs have the benefit of robust structure and the possible integration of pickup and modulation coils that allow operation of the SQUID at its highest sensitivity flux bias conditions using a flux -locked loop (FLL) feedback mechanism.

[005] The SOT, on the other hand, has the advantage of very small size, nanoscale proximity to the sample, and operation at high fields. The inability to use an FLL, however, poses a significant drawback as the high sensitivity of the SOT is achieved only at specific field values leaving substantial intermediate field regions as "blind spots".

[006] In conventional SQUID setups the magnetic field of the sample is not coupled to the SQUID loop directly but rather through a pickup coil. As a result, by adding a modulation coil or an integrated current-carrying element, the total flux in the SQUID loop can be maintained at its optimal bias while the sample can be exposed to any value of the applied magnetic field. The nanoscale proximity of the SOT to the sample surface, which is its key advantage, however, dictates that the flux in the SQUID loop cannot be varied without a corresponding invasive change in the local field of the sample. As a result, the FLL configuration cannot be used in direct nanoscale magnetic imaging.

SUMMARY

[007] Various embodiments of the present invention provide a novel multi -terminal multi -junction SQUID-On-Tip (mSOT) that allows sensitive continuous operation over

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a wide range of fields with spin sensitivity better than 8 μ _β Ηζ . In contrast to the FLL used in conventional SQUIDs, embodiments of the present invention feature a superconducting phase locked loop (SPLL) that allows in-situ electronic control of the SQUID's quantum interference pattern without the need to modify the flux in the SQUID loop. As a result, nanoscale magnetic imaging can be performed with optimal sensitivity at any value of the local field without "blind spots" and without affecting the local field. According to these embodiments, driving a current through a control terminal modifies the superconducting phase across one of the junctions, thus shifting the quantum interference pattern. A single control terminal allows shifting the pattern by half a period while two terminals enable a shift by a full period. As a result, various additional noise reduction schemes can be implemented analogous to the ones based on FLL protocols. An mSOT according to certain embodiments of the invention thus provide a new electronically-tunable tool for highly sensitive quantitative imaging and study of nanoscale magnetic phenomena over a large dynamic range.

[008] In particular, these embodiments provide a feedback mechanism that is advantageous to a variety of SQUIDs but is of crucial importance to SOTs. According to these embodiments, instead of controlling the magnetic flux in the SQUID loop, the phase of the superconducting order parameter in the loop is in-situ controlled. The resulting SPLL provides highly sensitive noninvasive SQUID operation at any value of the applied magnetic field.

[009] The basic principle of SPLL operation can be understood by considering an ideal SQUID with no inductance in its superconducting state,, which is described by:

( l + ψ2 + 2π(Φ _α /Φο) = 2πη (Equation 1)

[0010] where <¾ ^■ is the superconducting phase difference across each of the Josephson junctions (JJ), Φ _α is the applied flux in the loop, OQ is the flux quantum, and n is an integer. In the following we consider the situation of n = 0 for simplicity. The highest sensitivity of the SQUID is achieved when φι + >2 ~ π/2 corresponding to an applied flux of |Φ _α | « OQ/4. Since, at the critical current, and ς¾ also have to satisfy the requirement of maximum total supercurrent through the SQUID, there are no free degrees of freedom and the phases and of the two junctions at the critical current conditions are uniquely defined. Hence operation at the optimal sensitivity can be achieved only if the FLL maintains the net flux in the SQUID loop at |Φ _α | « OQ/4. [0011] Therefore, according to an embodiment of the present invention, there is provided a multi-terminal multi -junction Superconducting Quantum Interference Device On Tip (mSOT) apparatus including: (a) a multi-terminal multi -junction Superconducting Quantum Interference Device On Tip including: (b) a superconducting loop with a first Josephson Junction, a second Josephson Junction, and a third Josephson Junction; (c) a first terminal electrically connected to the loop between the third Josephson Junction and the first Josephson Junction; (d) a second terminal electrically connected to the loop between the first Josephson Junction and the second Josephson Junction; and (e) a third terminal electrically connected to the loop between the second Josephson Junction and the third Josephson Junction; and (f) a bias current source for applying a bias current to the first terminal; and (g) a first control current source for applying a first control current to the second terminal.

[0012] In addition, according to another embodiment of the present invention, there is also provided a method for operating a multi -terminal multi -junction Superconducting

Quantum Interference Device On Tip (mSOT) which includes: (a) a superconducting loop with a first Josephson Junction, a second Josephson Junction, and a third

Josephson Junction; (b) a first terminal electrically connected to the loop between the third Josephson Junction and the first Josephson Junction; (c) a second terminal electrically connected to the loop between the first Josephson Junction and the second

Josephson Junction; and (d) a third terminal electrically connected to the loop between the second Josephson Junction and the third Josephson Junction; (e) wherein the mSOT has a current-flux response function, and wherein the current-flux response function has a maximum absolute value; the method including: (f) applying a bias current to the first terminal; (g) applying a first control current through the second terminal; (h) and adjusting the first control current to minimize a flux noise of the mSOT for a given applied magnetic field.

BRIEF DESCRIPTION OF THE DRAWINGS

[0013] The subject matter disclosed may best be understood by reference to the following detailed description when read with the accompanying drawings in which: [0014] Fig. 1A is a schematic diagram of an mSOT showing a SQUID loop with four junctions and four terminals, according to an embodiment of the present invention. [0015] Fig. IB illustrates a cross section of the quartz tube with its four grooves and Pb film formed by two side depositions, according to an embodiment of the present invention.

[0016] Fig. 1C is a side-view scanning electron micrograph of a device according to an embodiment of the present invention.

[0017] Fig. ID is a top view scanning electron micrograph of the device shown in Fig. 1C.

[0018] Fig. 2A is a schematic diagram of an mSOT electrical circuit according to an embodiment of the present invention.

[0019] Fig. 2B illustrates mSOT electrical characteristics at various values of B _a , acquired at ¾2 = 74 μΑ and Ιβ^ = 0.

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[0020] Fig. 2C illustrates current noise spectral density S and corresponding flux

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noise at B _a = 0.2 T at the optimal working point. [0021] Fig. 3 A illustrates an interference pattern I _C \(B _A ) at three values of control current ¾2 = -150, -30, and 90 μΑ at Ι _β ^ = 0, according to an embodiment of the invention.

[0022] Fig. 3B is a plot of I _C \{IB2 _I -¾) showing a continuous shift of the interference pattern with the control current ¾2 at Ι _β ^ = 0.

[0023] Fig. 3C - Fig. 3E illustrate an mSOT current-flux response function, ά1\Ιά _ω for three values of Ι _β at Ι _β ^ = 0 showing the shift in field of the sensitive regions as the control current is varied.

[0024] Fig. 4 is a plot of flux noise and spin noise in an mSOT device according to an embodiment of the present invention, comparing the noise levels of the device with and without application of SPLL.

[0025] Fig. 5 is a flowchart illustrating a method according to an embodiment of the present invention.

[0026] Fig. 6 is a flowchart illustrating another method according to a related embodiment of the present invention.

[0027] Fig. 7 is a flowchart illustrating a further method according to another related embodiment of the present invention.

[0028] For simplicity and clarity of illustration, elements shown in the figures are not necessarily drawn to scale, and the dimensions of some elements may be exaggerated relative to other elements. In addition, reference numerals may be repeated among the figures to indicate corresponding or analogous elements. DETAILED DESCRIPTION

[0029] SPLL operation is based incorporating additional degrees of freedom by integrating additional junctions and terminals in the SQUID loop as shown schematically in Fig. 1A. According to certain embodiments of the present invention, there are at least three junctions and the number of terminals is equal to the number of junctions; i.e., every superconducting segment bordered by adjacent junctions is connected to its own terminal. In other embodiments, however, there are at least four junctions and at least three terminals, but not every segment has its own terminal.

[0030] Multi -junction configurations as well as multi terminal ones have been used in the past for logical circuits, enhanced SQUID capabilities, variable inductances, and flux modulation. In contrast, embodiments of the present invention use the additional degrees of freedom for phase control, rather than flux control, of the quantum interference pattern of the SQUID.

[0031] Fig. 1A shows a schematic configuration of an mSOT according to an embodiment of the invention. A superconducting loop 100 provides a circuit for current flow around an area 110 through which passes a magnetic flux Φ _α . A current I is applied to a first terminal 101 ("terminal 1"). A third terminal 103 ("terminal 3") functions as a current drain to ground, such that h = + + h. A second terminal 102 ("terminal 2") and a fourth terminal 104 ("terminal 4") are used to provide control currents I and I4, respectively. Loop 100 has four identical Josephson junctions: a first junction 111, a second junction 112, a third junction 113, and a fourth junction 114, through which currents Ji , J2, J3, and J4 circulate as indicated. For this structure Equation 1 is replaced by: Φΐ ⁺ Φ2 ⁺ Φ3 ⁺ Φ4 ⁺ 2π(Φ _α /Φο) = 2π ^η (Equation 2)

[0032] In a similar fashion to the case of a two-junction SQUID, the most sensitive SQUID response in the absence of control currents is achieved when Φΐ + Φ2 ⁺ Φ3 ⁺ Φ4 ^{¾ π} /2· The condition I = I = 0 dictates that φ\ = ς>2 and <¾ = 94, and hence φ\ + ς>4 = + Φ3 « π/4, restricting the optimal sensitivity operation to |Φ _α | « Φ ₀ /4 as in the case of a regular SQUID. However, by allowing control currents to flow through terminal 2 (102) and terminal 4 (104), this restriction is lifted as follows:

[0033] In the absence of control currents, the critical current Ι _€ \( _Α ) reaches its

JflQX

maximum value of I _C \ = 2JQ at zero applied flux Φ _Α = 0 where first junction 111 and fourth junction 114 each carry maximal dissipationless current Jg, inducing a phase drop of φι = -ψ4 = π/2. For simplicity current I4 is fixed at I4 = 0, thus imposing the condition that 93 = -94, while the control current I is given by I = J2 - Ji = Jo(sin ψ2 - sin φι ) = Jo(sin - 1). For any applied flux Φ _α , Φΐ = -Φ4 = -Φ3 = π/2 is kept fixed and ς>2 is modified by adjusting I to satisfy Equation 2. Since ( i and 94 are not changed, the current I of the mSOT remains at its maximal critical value as previously specified, namely

where the control current I required to keep I _c \ at its maximum is given by Equation 3 :

I ₂ = Jo(sin ς>2 - 1) = Jo(sin (π/2 - 2πΦ _α /Φ ₀ ) - 1)

J ₀ (cos (2πΦ _α /Φ ₀ ) - 1) (Equation 3)

[0034] Thus, according to various embodiments of the present invention the quantum interference pattern Ι _€ \(Φ _Α ) of the mSOT is shifted as a whole via electronic means by applying control currents and I4. In particular, according to embodiments of the present invention, for any value of the applied flux Φ _α a control current exists that biases the mSOT to the most sensitive working point. According to a related embodiment, this control is applicable to any SQUID with at least three junctions and three terminals including asymmetric junctions, non-sinusoidal current-phase relations, and the presence of finite inductance. In contrast to the previously-known FLL, by which the interference pattern is shifted by applying additional flux to the SQUID loop, SPLL embodiments of the present invention, as disclosed herein, achieve a shift of the pattern by controlling the superconducting phase across one (or more) additional junctions in the loop. As a result, a very sensitive measurement of the local magnetic field of a sample can be attained at any value of the local field thus eliminating the "blind spots". Note also, that due to its very small size the geometrical inductance of the SOT is several orders of magnitude smaller than its kinetic inductance. As a result, the control current affects the superconducting phase across the junctions with negligible change to the self-induced flux in the SQUID loop.

[0035] A two-terminal SOT may be fabricated by self-aligned two-sided deposition of two superconducting leads along a pulled quartz pipette followed by a third deposition on the apex ring. The principal challenge in fabricating the mSOT is in attaining nanoscale multi-terminal connections to the apex.

[0036] According to particular embodiments of the invention, fabrication of an mSOT begins with a quartz tube or pipette having an outer diameter of the order of 1 mm, and

2

having four 0.1 x 0.15 mm grooves equally spaced on its outer circumference, which maintain their relative shape upon laser heating and pulling, and which extend to an apex formed during the pulling. Fig. IB illustrates a cross-section of the finished mSOT according to a related embodiment of the present invention, and Figs. 1C and ID are scanning electron microscope (SEM) images of the mSOT exterior and tip. The grooves provide shading during two side depositions of lead (Pb) to create four gaps 171, 172,

173, and 174, respectively associated with the grooves, and separating four leads 153, 154, 155, and 156. One side Pb deposition is made in a direction 161, and the other side deposition is made in a direction 162. Quartz body 151 has a hollow interior 152 which extends the full length of the mSOT to an apex ring 181. In a third Pb deposition on apex ring 181, four constrictions are formed in the regions of gaps 171, 172, 173, and

174, creating Dayem-bridge weak links to establish a self-aligned four-junction four- terminal mSOT. Other embodiments of the invention utilize other superconducting materials, non-limiting examples of which include Indium (In), Niobium (Nb), and combinations of different materials, such as Pb-In. Still further embodiments of the invention provide other fabrication techniques, such as lithographic techniques, for producing mSOTs.

[0037] According to a related embodiment, an mSOT is characterized at 4.2 K using the electrical circuit shown in Fig. 2A, where a current bias Ι _β \ from a current source 211 is applied to the mSOT in parallel with a shunt resistor 221 The resulting current I\ flowing into the mSOT is measured using a series SQUID array amplifier (SSAA) 201. The control currents I and I4 are provided by applying to the mSOT control bias currents Ι _β from a current source 212 and Ι _β ^ from a current source 214, respectively, in parallel with a shunt resistor 222 and a shunt resistor 224, respectively. In a related embodiment, the control terminals of the mSOT are current biased, and thus shunt resistors 222 and 224 are unnecessary. Fig. 2B illustrates electrical characteristics I\ (Ι _β \ ) of the mSOT device showing the measured I\ versus the applied bias current Ι _β at several values of the applied field B _a . For Ι\ ^< Ι _α \{Β _α ) the mSOT is in the superconducting state and most of the applied current Ιβ\ flows through the device. When I reaches I _c i(B _a ) the mSOT becomes resistive and a significant part of the current is diverted to shunt resistor 201 At higher biases a number of resonances are visible.

[0038] To illustrate the electronic tunability of an mSOT according to an embodiment of the present invention, Fig. 3A shows an interference pattern of the critical current I _c l(B _a ) derived from electrical characteristics, in a curve 301 for ¾ = 90 μΑ, a curve

302 for Ι _β 2 = -30 μΑ, and a curve 303 for Ι _β ΐ = -150 μΑ. An interference period 304 of

97 mT corresponding to an effective loop diameter of 165 nm agrees with the dimensions shown in the SEM image of Fig. ID. According to embodiments of the present invention, a noteworthy feature illustrated by Fig. 3A is the in-situ electronic tunability of the mSOT. The three presented I _c i(B _a ) curves 301, 302, and 303 show that the entire pattern is readily shifted horizontally by varying the control bias current Ι _β ^ (keeping Ιβ^ = 0 for simplicity). Fig. 3B shows that the critical current interference pattern I _c \(B _a , Ιβ ) is shifted continuously with Ιβΐ over a large span of about half a period between the lowest and highest presented values of Ιβΐ- This implies that for any value of B _a there is a current Ι _β ^ for which maximum mSOT sensitivity can be attained at this field, thus allowing optimal operation at any B _a without "blind spots". [0039] The first two terms of Equation 3 can be rewritten as Ι α) ⁼ -¾( 2) " Ά) _>

J lQX

namely curves 305, 306, and 307 tracing the maximum critical current I _c in Fig. 3B depict the current-phase relation J2(( 2) of junction 2. This correspondence holds for any form of current-phase relations and therefore provides a general tool for direct probing of current-phase relations in various systems.

[0040] Achieving highly sensitive continuous operation with low flux noise

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¾> ⁼ $1 /| άΙ\ΙάΦ _α I requires a high response {i.e., derivative of current with respect

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to flux I άΙ\ΙάΦ _α I) and a low current noise S at all values of the applied field. Fig. 3C shows the measured current-flux response function άΙ\ΙάΦ _α versus bias current Ι \ and the applied field B _a for a bias current of ¾ = 90 μΑ. The blue and red regions have high values of | άΙ ΙάΦ _α | whereas the green regions have no sensitivity due to either the condition that According to embodiments of the present invention, varying ¾2 shifts the sensitive regions to the desired value of B _a as shown in Figs. 3D and 3E.

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[0041] A spectrum of the current noise S of the mSOT is shown in Fig. 2C, displaying a l/ ^" behavior at low frequencies ≤ 1 kHz followed by white noise at higher frequencies.

[0042] By measuring the white noise level and the current-flux response function

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άΙγΙάΦ _α , the flux noise = S Ι\άΙγΙάΦ _α \ of the mSOT versus B _a is derived. The

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spin noise S _n in units of μ _β /Ηζ is related to the flux noise through

„ 1/2 „ 1/2 , , ₀ 1/2 . . . „ _{Λ ηΎ} 111 „ -15 . ,

⁼ ^Φ ^r ' ^r ei where ύφ is in units or OQ/HZ ; r _e = 2.82 x 10 m is the classical electron radius; and r = 82 nm is the effective radius of the mSOT. To minimize the noise, the absolute value of the current-flux response function | άΙ ΙάΦ _α \ is maximized. Absolute value is taken, because only the magnitude of the quantity is significant - the sign does not matter. In descriptions herein of embodiments of the present invention, the term "current-flux response function" denotes the absolute value I άΙγΙάΦ _α \.

[0043] Fig. 4 shows both flux noise (left axis) and spin noise (right axis) of an mSOT device. A curve 401 with units on the left axis displays the flux noise measured at ¾2 = ¾4 = 0 {i.e., no control current), reflecting conventional two-terminal SOT operation. Curve 401 shows high noise levels in the "blind spot" regions - the large peaks in the flux noise of curve 401 correspond to regions where | άΙ ΙάΦ _α | is small.

[0044] In contrast, a curve 402 shows the performance of the same mSOT device operating in the SPLL mode by adjusting the control currents according to embodiments of the present invention. For each value of B _a , optimal values of the control currents ¾2 and ¾4 are determined, which bias the mSOT to its optimal working point and thereby minimize the flux noise. The additional degrees of freedom provided by the control currents result in the elimination of "blind spots" and a uniform and low noise of 5 to 8

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μβ/Ηζ over a previously unattainable continuous range of applied fields from 0 to 0.5 T.

[0045] In various embodiments of the present invention, therefore, the performance of

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the mSOT is optimized by minimizing the flux noise for a given magnetic field. In

1/2 a particular embodiment, this is done by minimizing the current noise S _j as well as maximizing the current-flux response function | άΙ ΙάΦ _α |; in a related embodiment, the

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current noise S is determined by other considerations, and only the current-flux response function | άΙ\ΙάΦ _α | is maximized for a given magnetic field.

[0046] For the following methods, an mSOT characterization measurement of the current through terminal 101 (terminal 1) (B _a , I _B\ , hi, h< _\ ) is performed. From this, derive hi, /B ₄ ). Then magnetic field B _a may be inferred from bias current h and control currents I _B\ , hi, and 7g ₄ by adjusting them to obtain optimal flux noise levels for B _a .

[0047] Fig. 5 is a flowchart illustrating a method according to an embodiment of the present invention for operating an mSOT 500 having three terminals 501, 502, and 503; and three junctions 511, 512, and 513. In a step 521 a bias current is applied through terminal 501, and in a step 522 a value of current-flux response function | άΙ ΙάΦ _α | is obtained. Next, in a step 523, a control current I ₂ is applied through terminal 502. In a step 524 a maximum value of the current-flux response function, | άΙ\ΙάΦ _α \ ^max , is obtained. Finally, in a step 525, control current I ₂ is adjusted to maximize the current- flux response function | άΙ ΙάΦ _α |, so that | άΙ ΙάΦ _α | is substantially at the maximum value I άΙ _λ ΙάΦ _α \ ^max (i.e., so that | άΙ _λ ΙάΦ _α \≡ \ άΙ _λ ΙάΦ _α \ ^max ).

[0048] Fig. 6 is a flowchart illustrating another method according to a related embodiment of the present invention for operating an mSOT 600 having four terminals 601, 602, 603, and 604; and four junctions 611, 612, 613, and 614. In a step 621 a bias current is applied through terminal 601, and in a step 622 a value of current-flux response function | άΙ\ΙάΦ _α \ is obtained. Next, in a step 623, a first control current I ₂ is applied through terminal 602. In a step 624 a second control current 7 ₄ is applied through terminal 604. Then, in a step 625, a maximum value of the current-flux response function, | άΙ\ΙάΦ _α \ ^max , is obtained. Finally, in a step 626, first control current and second control current 7 ₄ are adjusted to maximize the current-flux response function | άΙ ΙάΦ _α |, so that | άΙ ΙάΦ _α | is substantially at the maximum value

I άΙ _γ ΙάΦ _α \ ^max (i.e., so that | άΙ _γ ΙάΦ _α \≡ \ άΙ _γ ΙάΦ _α \ ^max ).

[0049] Fig. 7 illustrates a further method according to another related embodiment of the present invention, which can be carried out to augment the methods disclosed above. In a step 701, the mSOT is characterized according to measurements of the current (B _a , I _B hi, I _B ) and the current noise hi, I _B ) as described previously, to

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obtain a flux noise characterization 711 φ = S l\ άΙγΙάΦ _α |. Next, in a step 702, a set 712 of mSOT working regions is obtained. A working region is specified by a rough value of for mSOT operation. Then, in a step 703 the bias of the mSOT is adjusted for a working region selected from set 712. In a related embodiment, the selection of working region is arbitrary. Following this, in a step 704 is measured and an estimated value 713, B _a ^es is inferred from the measurement of according to characterization 711. Because value 713 is only an estimate, in a step 705, the mSOT is fine-tuned to an optimal working point of minimal noise. Finally, in a step 706 I _\ for the optimal working point is measured, and a measured value 714, B _a ^mea , is inferred from the measurement of according to characterization 711.